There is a verb that sounds like kwel/quel that means "to calm". My best attempts to get a butchered spelling close enough that a digital dictionary can tell me the proper spelling are failures; and none of my thesauri have the word of which I'm thinking.

Can anybody help out a frustrated and spelling-challenged English speaker?

The word is quell. You'd certainly have found it by googling or dictionary-searching if it weren't for the fact that the acronym QUEL is all that comes up when you try to google "quel" or search a dictionary for it.